Question

factor the given polynomial by finding the greatest common monomial factor (or the negative of the greatest common monomial factor) and rewrite the expression.

\\10xy + 2xy^2 + 5x^3y\\

Explanation:

Identify the terms of the polynomial

We begin with the given polynomial:

$$ 10xy + 2xy^2 + 5x^3y $$

This expression consists of three distinct monomial terms.

Find the greatest common factor of the coefficients

Using the Greatest Common Factor knowledge point, we find the greatest common divisor (GCD) of the numerical coefficients:

• The coefficients are \(10\), \(2\), and \(5\).
• The only positive integer that divides \(10\), \(2\), and \(5\) is \(1\).
• Thus, the numerical part of the greatest common factor is \(1\).

Find the greatest common factor of the variable parts

Using the Factoring Monomials knowledge point, we determine the lowest power of each variable present in all terms:

• For \(x\): the terms have \(x^1\), \(x^1\), and \(x^3\). The lowest power is \(x^1 = x\).
• For \(y\): the terms have \(y^1\), \(y^2\), and \(y^1\). The lowest power is \(y^1 = y\).
• Combining these, the greatest common monomial factor (GCMF) is \(xy\).

Divide each term by the common factor

We divide each term of the polynomial by the GCMF, \(xy\), to find the remaining terms:

• First term: \(\frac{10xy}{xy} = 10\)
• Second term: \(\frac{2xy^2}{xy} = 2y\)
• Third term: \(\frac{5x^3y}{xy} = 5x^2\)

Rewrite the factored expression

We write the polynomial as the product of the GCMF and the sum of the remaining terms:

$$ xy(10 + 2y + 5x^2) $$

Answer:

\(xy(10 + 2y + 5x^2)\)